1. On the Marginal Hilbert Spectrum

2. Outline Definitions of the Hilbert Spectra (HS) and the Marginal Hilbert Spectra (MHS). Computation of MHS The relation between MHS and Fourier Spectrum MHS with different frequency resolutions Examples

3. Hilbert Spectrum

4. Definition of Hilbert Spectra

5. Hilbert Spectra

6. Definition of the Marginal Hilbert Spectrum

7. can be amplitude or the square of amplitude (energy). d ω d t Schematic of Hilbert Spectrum

8. Computing Hilbert Spectrum

9. Marginal Spectrum

10. Hilbert and Marginal Spectra

11. Some Properties

12. MHS and Fourier Spectra

13. MHS with different Resolutions

14. Some Properties The spectral density depends on the bin size that is on both temporal and frequency resolutions. For marginal Frequency spectrum, the temporal resolution is implicit. For instantaneous energy density, the frequency resolution is not implicit. Frequency assumes instantaneous value, not mean; it is not limited by the Nyquist. We can zoom the spectrum to any temporal and frequency location.

15. Fourier vs. Hilbert Spectra Adaptive basis, Data Driven Time-frequency spectrum Physical meaningful frequency at both the high and low frequency ranges Resolution of the frequency adjustable Zoom capability Marginal spectra for frequency and time.

16. Example Delta-Function

18. Influence of the resolution of frequency on the Hilbert-Huang spectrum [ 10 20 50 100 300 500 600 800]/1000

19. Effects of Frequency Resolution

20. Fourier Energy Spectrum

21. Example Uniformly distributed white noise

22. Data

23. Data : IMF

24. Fourier Spectra

26. Hilbert Spectra : Various F-Resolutions

27. Hilbert Spectra : Various T-Resolutions

28. Hilbert Amplitude Spectra : Various F-Resolutions

29. STD = 0.2 Data : White Noise STD = 0.2

30. Fourier Power Spectrum

31. IMF

32. Hilbert Marginal and Fourier Spectra

33. Factor = 1 Effects on Frequency Resolution MHS

34. Normalized MHS

35. [ 10 20 50 100 300 500 600 800]/1000

36. Effect Frequency Resolution : bin size

37. Normalized

38. Example Earthquake data

39. Earthquake data E921

40. IMF EEMD2(3,0.2,100)

41. IMF EEMD2(3,0.1,10)

42. IMF EEMD2(3,0,1)

43. Different Frequency Resolutions VS Fourier and Normalization

44. MHS and Fourier at full resolutions

45. MHS and Fourier Normalized

46. MHS Smoothed and Normalized

47. MHS Different Frequency Resolutions

48. MHS Different Resolutions Normalized

49. MHS EMD and EEMD

50. Zoom

51. MHS 100 Ensemble

53. MH Amplitude Spectrum

54. 10 Ensemble Poor normalization

55. Fourier and Hilbert Marginal Spectra

56. Normalized

57. Effect of Filter size : Fourier

58. Hilbert Spectrum

62. Marginal Spectra

63. Normalized

64. Zoom Effects

65. Normalized

66. Effect of bin size

67. Normalized

68. Effects of bin size and zoom

69. Normalized

70. Summary Hilbert spectra are time-frequency presentations. The marginal spectra could have various resolutions and zoom capability. Hilbert marginal spectra could be smoothed without losing resolution. Another marginal Hilbert quantity is the time- energy distribution.

71. Summary For long time, the Hilbert Marginal Spectrum was not defined absolutely. The energy and amplitude spectra were not clearly compared; they are totally different spectra. Clear conversion factor are given to make comparisons between MHS and Fourier easily. Conversion factor also was provided for MHS with different Frequency resolutions. In,most cases the MHS in energy is very similar to Fourier, for the temporal has been integrated out.